
theorem Th95:
  for G1 being _Graph, G2 being DGraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 holds
    v2.inNeighbors() = the_Vertices_of G2 \ (v1.inNeighbors() \/ {v2}) &
    v2.outNeighbors() = the_Vertices_of G2 \ (v1.outNeighbors() \/ {v2})
proof
  let G1 be _Graph, G2 be DGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  consider G9 being DLGraphComplement of G1 such that
    A2: G2 is removeLoops of G9 by Def8;
  reconsider v9 = v1 as Vertex of G9 by Def6;
  thus v2.inNeighbors() = v9.inNeighbors() \ {v2} by A1, A2, GLIBPRE0:63
    .= (the_Vertices_of G9 \ v1.inNeighbors()) \ {v2} by Th60
    .= the_Vertices_of G9 \ (v1.inNeighbors() \/ {v2}) by XBOOLE_1:41
    .= the_Vertices_of G2 \ (v1.inNeighbors() \/ {v2}) by A2, GLIB_000:53;
  thus v2.outNeighbors() = v9.outNeighbors() \ {v2} by A1, A2, GLIBPRE0:63
    .= (the_Vertices_of G9 \ v1.outNeighbors()) \ {v2} by Th60
    .= the_Vertices_of G9 \ (v1.outNeighbors() \/ {v2}) by XBOOLE_1:41
    .= the_Vertices_of G2 \ (v1.outNeighbors() \/ {v2}) by A2, GLIB_000:53;
end;
